Stochastic and variational approach to finite difference approximation of Hamilton-Jacobi equations

TL;DR

This paper extends a stochastic variational finite difference scheme for Hamilton-Jacobi equations to higher dimensions, demonstrating stability, convergence, and approximation of solutions and characteristics using probabilistic methods.

Contribution

It generalizes previous one-dimensional results to multiple dimensions, introducing a deterministic scheme with stochastic calculus of variations for better approximation and analysis.

Findings

Proves stability and convergence of the scheme in higher dimensions

Provides approximation of viscosity solutions and characteristic curves

Establishes convergence rates using probability theory

Abstract

The author presented a stochastic and variational approach to the Lax-Friedrichs finite difference scheme applied to hyperbolic scalar conservation laws and the corresponding Hamilton-Jacobi equations with convex and superlinear Hamiltonians in the one-dimensional periodic setting, showing new results on the stability and convergence of the scheme [Soga, Math. Comp. (2015)]. In the current paper, we extend these results to the higher dimensional setting. Our framework with a deterministic scheme provides approximation of viscosity solutions of Hamilton-Jacobi equations, their spatial derivatives and the backward characteristic curves at the same time, within an arbitrary time interval. The proof is based on stochastic calculus of variations with random walks; a priori boundedness of minimizers of the variational problems that verifies a CFL type stability condition; the law of large numbers for random walks under the hyperbolic scaling limit. Convergence of approximation and the rate of convergence are obtained in terms of probability theory. The idea is reminiscent of the stochastic and variational approach to the vanishing viscosity method introduced in [Fleming, J. Differ. Eqs (1969)].